Input Sparsity and Hardness for Robust Subspace Approximation

In the subspace approximation problem, we seek a k-dimensional subspace F of R^d that minimizes the sum of p-th powers of Euclidean distances to a given set of n points a_1, ..., a_n in R^d, for p >= 1. More generally than minimizing sum_i dist(a_i,F)^p,we may wish to minimize sum_i M(dist(a_i,F)) for some loss function M(), for example, M-Estimators, which include the Huber and Tukey loss functions. Such subspaces provide alternatives to the singular value decomposition (SVD), which is the p=2 case, finding such an F that minimizes the sum of squares of distances. For p in [1,2), and for typical M-Estimators, the minimizing $F$ gives a solution that is more robust to outliers than that provided by the SVD. We give several algorithmic and hardness results for these robust subspace approximation problems. We think of the n points as forming an n x d matrix A, and letting nnz(A) denote the number of non-zero entries of A. Our results hold for p in [1,2). We use poly(n) to denote n^{O(1)} as n -> infty. We obtain: (1) For minimizing sum_i dist(a_i,F)^p, we give an algorithm running in O(nnz(A) + (n+d)poly(k/eps) + exp(poly(k/eps))), (2) we show that the problem of minimizing sum_i dist(a_i, F)^p is NP-hard, even to output a (1+1/poly(d))-approximation, answering a question of Kannan and Vempala, and complementing prior results which held for p >2, (3) For loss functions for a wide class of M-Estimators, we give a problem-size reduction: for a parameter K=(log n)^{O(log k)}, our reduction takes O(nnz(A) log n + (n+d) poly(K/eps)) time to reduce the problem to a constrained version involving matrices whose dimensions are poly(K eps^{-1} log n). We also give bicriteria solutions, (4) Our techniques lead to the first O(nnz(A) + poly(d/eps)) time algorithms for (1+eps)-approximate regression for a wide class of convex M-Estimators.

Comments: paper appeared in FOCS, 2015

Similar Publications

Randomized binary exponential backoff (BEB) is a popular algorithm for coordinating access to a shared channel. With an operational history exceeding four decades, BEB is currently an important component of several wireless standards. Despite this track record, prior theoretical results indicate that under bursty traffic (1) BEB yields poor makespan and (2) superior algorithms are possible. Read More


An $(r, \ell)$-partition of a graph $G$ is a partition of its vertex set into $r$ independent sets and $\ell$ cliques. A graph is $(r, \ell)$ if it admits an $(r, \ell)$-partition. A graph is well-covered if every maximal independent set is also maximum. Read More


Triangle-free graphs play a central role in graph theory, and triangle detection (or triangle finding) as well as triangle enumeration (triangle listing) play central roles in the field of graph algorithms. In distributed computing, algorithms with sublinear round complexity for triangle finding and listing have recently been developed in the powerful CONGEST clique model, where communication is allowed between any two nodes of the network. In this paper we present the first algorithms with sublinear complexity for triangle finding and triangle listing in the standard CONGEST model, where the communication topology is the same as the topology of the network. Read More


This paper formulates a novel problem on graphs: find the minimal subset of edges in a fully connected graph, such that the resulting graph contains all spanning trees for a set of specifed sub-graphs. This formulation is motivated by an un-supervised grammar induction problem from computational linguistics. We present a reduction to some known problems and algorithms from graph theory, provide computational complexity results, and describe an approximation algorithm. Read More


In this work, we provide a general framework for adding a linearizable iterator to data structures with set operations. We propose a condition on these set operations, called locality, so that any data structure implemented from local atomic operations can be augmented with a linearizable iterator as described by our framework. We then apply the iterator framework to various data structures, prove locality of their operations, and demonstrate that the iterator framework does not significantly affect the performance of concurrent operations. Read More


In the k-partition problem (k-PP), one is given an edge-weighted undirected graph, and one must partition the node set into at most k subsets, in order to minimise (or maximise) the total weight of the edges that have their end-nodes in the same cluster. Various hierarchical variants of this problem have been studied in the context of data mining. We consider a 'two-level' variant that arises in mobile wireless communications. Read More


We introduce a technique to turn dynamic programming tasks on labeled directed acyclic graphs (labeled DAGs) closer to their sequence analogies. We demonstrate the technique on three problems: longest increasing subsequence, longest common subsequence, and co-linear chaining extended to labeled DAGs. For the former we obtain an algorithm with running time $O(|E| k \log |V|+ |V| k \log^2 |V|)$, where $V$ and $E$ are the set of vertices and edges, respectively, and $k$ is the minimum size of a path cover of $V$. Read More


We show that Boolean matrix multiplication, computed as a sum of products of column vectors with row vectors, is essentially the same as Warshall's algorithm for computing the transitive closure matrix of a graph from its adjacency matrix. Warshall's algorithm can be generalized to Floyd's algorithm for computing the distance matrix of a graph with weighted edges. We will generalize Boolean matrices in the same way, keeping matrix multiplication essentially equivalent to the Floyd-Warshall algorithm. Read More


Integer Linear Programming is a famous NP-complete problem. Lenstra showed that in the case of small dimension, it can be solved in polynomial time. This algorithm became a ubiquitous tool, especially in the design of parameterized algorithms for NP-complete problems, where we wish to isolate the hardness of an instance to some parameter. Read More


Given a string $T$, it is known that its suffix tree can be represented using the compact directed acyclic word graph (CDAWG) with $e_T$ arcs, taking overall $O(e_T+e_{{\overline{T}}})$ words of space, where ${\overline{T}}$ is the reverse of $T$, and supporting some key operations in time between $O(1)$ and $O(\log{\log{n}})$ in the worst case. This representation is especially appealing for highly repetitive strings, like collections of similar genomes or of version-controlled documents, in which $e_T$ grows sublinearly in the length of $T$ in practice. In this paper we augment such representation, supporting a number of additional queries in worst-case time between $O(1)$ and $O(\log{n})$ in the RAM model, without increasing space complexity asymptotically. Read More